# How do you find a power series representation for  f(x)= 9/(1-3x)^2  and what is the radius of convergence?

Feb 16, 2018

See below

#### Explanation:

$\frac{3}{1 - 3 x} ^ 2 = \frac{d}{\mathrm{dx}} \left(\frac{1}{1 - 3 x}\right)$ but for $\left\mid 3 x \right\mid < 1$

$\frac{1}{1 - 3 x} = {\lim}_{n \to \infty} {\sum}_{k = 0}^{n} {3}^{k} {x}^{k}$ hence

$f \left(x\right) = \frac{9}{1 - 3 x} ^ 2 = 3 {\sum}_{k = 0}^{\infty} {3}^{k} {x}^{k}$

convergent for $\left\mid x \right\mid < \frac{1}{3}$